U N I V E R S I TÀ D I P I S A d i pa r t i m e n t o d i f i s i c a e. fermi l au r e a m a g i s t r a l e i n f i s i c a T E S I D I L A U R E A M A G I S T R A L E
R E N O R M A L I Z A B I L I T Y A N D
F I N I T E N E S S O F N O N L O C A L
Q U A N T U M G R AV I T Y
c a n d i d at o r e l at o r e S T E FA N O L A N Z A P R O F. D A M I A N O A N S E L M I A N N O A C C A D E M I C O 2 0 1 5 / 2 0 1 6Stefano Lanza: Renormalizability and finiteness of nonlocal quantum gravity, c July 2016
P R E FA C E
Quantum mechanics and general relativity are the two main heritages of the last century physics. Both theories have solid experimental foundations and are equally valid in their own fields of application. On the one hand, general relativity has explained phenomena like the precession of Mercury and predicted others long before they were experimentally seen, from black holes to the very recent verification of the existence of the gravitational waves. In sum, it explained the structure of the universe as a whole and the rules among macroscopic objects. On the other hand, the realm of quantum mechanics is the infinitesimally small universe: it explains the interactions among particles through quantum field theory and helped to predict the ex-istence of new particles (the Higgs boson is one of the last examples). In the microscopic world Einstein’s theory does not play any relevant role, since its effects are negligible, allowing to consider flat space as an acceptable ap-proximation.
When quantum field theory was developed in the second half of the twen-tieth century, it was natural to promote spacetime to a curved entity, in order to include general relativity in the quantum framework. It was sud-denly clear that such an approach was unsatisfactory, since, due to many serious problems, quantum field theory was spoiled of one of its impor-tant features, the renormalizability. During the last decades, many different solutions have been proposed: some of them, like string theory and its fur-ther developments, introduce radical physical changes; ofur-ther approaches are softer in the sense that they try to be as coherent with quantum field theory as possible. In this work we will explore one of the second possibilities, the nonlocal theory of quantum gravity that has recently aroused interest among some groups of theoretical physicists.
Our aim is to correctly incorporate gravity in quantum field theory, in order to handle it with the powerful tools that are extensively and successfully used in particle physics. The guiding principles designing the nonlocal the-ory are the following:
• Perturbative renormalizability: interactions can be depicted as an infinite series of Feynman diagrams, some of which can generate divergences, conveniently parametrized. The renormalization tells us how to con-trol the infinities of the theory: in order to eliminate them, we must add counterterms to the bare Lagrangian, that have to be of a finite number of types to achieve renormalizability.
• Lorentz invariance: it is a well tested symmetry of nature and it should be a symmetry of quantum gravity as well.
• Unitarity: that is, calling S the scattering matrix of some process, the condition SS†= 1must be satisfied. Otherwise, the theory cannot be considered a faithful description of reality, since some pieces would actually be missing.
The only principle we decide to break is locality: we assume that the La-grangian can contain terms that are nonpolynomial in the derivatives. This work is organized into two parts: the first two chapters draw a path
that leads from Einstein’s gravity to nonlocal gravity, while the last three chapters constitute a study of the main aspects of nonlocal gravity; in detail: 1. in the first chapter we start with a brief summary of Einstein’s theory of gravitation, after which we proceed with the usual quantization of the classical theory: Einstein’s gravitational Lagrangian is the starting point of the quantum theory, with a flat background and a particle -the graviton - describing -the properties of -the curved spacetime. What we obtain is a unitary theory which is not renormalizable, due to the presence of a coupling constant with negative dimension in units of mass;
2. in the second chapter we look at the solutions to the renormalization problem of quantum gravity. A first attempt was made in 1977 by Stelle [19], who presented a higher derivative version of Einstein’s
the-ory: the renormalizability is restored, with the price of losing unitar-ity. We are naturally directed towards a nonlocal theory of gravity by introducing nonpolynomial expressions in the bare gravitational La-grangian. We examine Kuz’min’s nonlocal theory of gravity, that has been amended with some additional hypotheses in the last few years; 3. in the third chapter we systematically study the principles of unitarity and causality as basic ingredients to build a field theory. We see in which sense local gauge theories are unitary and causal and we would like these features to be present in the nonlocal theory as well. We see that it is not simple to show that nonlocality is compatible with them and we do not give a final answer, since the research is still ongoing; 4. in the fourth chapter, we review the observations made in [24] and [8],
in order to show a simple way to rewrite the nonlocal vertices. Then we present a unique and original complete demonstration of the super-renormalizability of the nonlocal theory of quantum gravity. We arrive at the conclusion that actually only four parameters are subjected to renormalization and, with a suitable choice of the nonlocal functions, only one loop diagrams diverge and the renormalization procedure does not even produce nonlocal counterterms;
5. finally, in the last chapter, in connection with some recent develop-ments in [13] and [14], we add another requirement to Kuz’min’s
the-ory: the finiteness, that is achieved by adding a potential at least cubic in Riemann tensor to the nonlocal gravitational Lagrangian. We com-pute explicitly the divergences for a minimal correction that is quartic in the graviton field; then, we propose an original, general method of dealing with corrections that are quartic in Riemann tensor.
C O N T E N T S
1 f r o m c l a s s i c a l t o q ua n t u m g r av i t y 1 1.1 Describing a curved spacetime 1
1.2 The Einstein action 3
1.3 The graviton 4
1.4 Gauge fixing in the canonical formalism 7 1.4.1 Canonical formalism 8
1.4.2 Canonical transformations 10 1.4.3 Gauge fixing 11
1.5 Quantum gravity: gravity as a gauge theory 13 1.6 Divergences of quantum gravity 16
2 t h e p r o b l e m o f r e n o r m a l i z a b i l i t y 21
2.1 Stelle’s theory: higher derivative quantum gravity 21 2.2 The loss of unitarity in higher derivative models 22 2.3 Kuz’min’s theory: nonlocal quantum gravity 24 2.4 Some properties of entire functions 25
2.5 Hypotheses for the nonlocal function 26 2.6 An explicit example of nonlocal function 28 2.7 The exponential function 31
3 s y s t e m at i c s o f u n i ta r i t y a n d c au s a l i t y 33 3.1 Unitarity and causality as basic requirements 33
3.1.1 Unitarity condition 33
3.1.2 Bogoliubov’s causality condition 34 3.2 Building up a new formalism 36
3.3 Cutting rules 40
3.4 The largest time equation and its consequences 40 3.4.1 Unitarity 42
3.4.2 Causality 43
3.5 Summing over intermediate states 45 3.6 The case of non local theories 48 4 v e r t i c e s a n d p o w e r c o u n t i n g 51
4.1 Vertices of nonlocal gravity 51 4.2 Properties ofS
l(z) 54
4.3 UV behavior of non local vertices 58 4.4 UV behavior of diagrams 59
4.5 Structure of counterterms 61 4.5.1 Locality of counterterms 61
4.5.2 Correspondence between nonlocal and higher deriva-tive theories 63
4.5.3 The renormalized Lagrangian of nonlocal quantum
grav-ity 63
5 f i n i t e n o n l o c a l g r av i t y 65
5.1 In search for a finite theory of quantum gravity 65 5.2 Influences of the new terms on renormalization 66
5.2.1 Renormalization of the curvature and the cosmologi-cal constant 67
5.2.2 Renormalization of the quadratic operators 68 5.3 An explicit example 70
5.4 The general case 73
vi c o n t e n t s
a n o tat i o n a n d u s e f u l e x pa n s i o n s 79 a.1 Notation and definitions 79
a.2 Useful expansions 80 b r e n o r m a l i z at i o n 81
b.1 Γ function and solid angle in D dimensions 81
b.2 Convergence 82
b.3 Feynman parameters and other useful integrals 82 b i b l i o g r a p h y 83
1
F R O M C L A S S I C A L T O
Q U A N T U M G R AV I T Y
1.1 d e s c r i b i n g a c u r v e d s pa c e t i m e
Spacetime is a four dimensional continuum, described by a set of coordi-nates xµ, where the 0-th component stands for time and the other three, that we will denote as xi, are the space components. General relativity broke down the Newtonian prejudice to think of spacetime as a flat entity, intro-ducing the idea that is better explained by a manifold, which, in general, is curved and resembles flat spacetime only locally.
Explicitly, if we stick with a spacetime point P of the manifold, we can de-fine the tangent space VP to the manifold in P, which is a four dimensional
vector space as well. We can define a basis{eµ}, so that a vector v in VP can
be written as:
v = vµeµ (1.1)
where vµare the vector components and with the caveat that{e
µ} has only
a local meaning.
Choosing different coordinates x0µ, the new vector components are related to the old ones by the transformation rule:
v0µ= ∂x
0µ
∂xν v
ν (1.2)
The vector defined in Eq.1.1is however a coordinate independent object and the basis vectors transform as:
e0µ=
∂xν
∂x0µeν (1.3)
For instance, the partial derivatives ∂µrepresent a good basis for VP.
The cotangent space VP∗ is defined as the space of linear maps f : VP→ R; its
basis vectors ωµare defined by their action over e µ:
ωµ(eν)≡ δµν (1.4)
A covector α in VP∗can then be represented as:
α = αµωµ (1.5)
The components of a covector αµand the basis ωµtransform according to:
αµ0 = ∂x ν ∂x0µαν, ω 0µ= ∂x0µ ∂xν ω ν (1.6)
The differentials dxµare an example of basis covectors.
In general, we can define a (l, k) type tensor T : T = Tµ1...µl
ν1...νkeµ1⊗ . . . ⊗ eµl⊗ ω
ν1⊗ . . . ⊗ ωνk (1.7)
2 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
whose components transform as:
T0µ10...µl0 ν10...νk0 = l Y i=1 ∂x0µi0 ∂xµi k Y j=1 ∂xνj ∂x0νj0 Tµ1...µl ν1...νk (1.8) The metric g = gµνdxµ⊗ dxν (1.9)
is a (0, 2) type tensor, that is:
• symmetric: ∀ v1, v2∈ VPg(v1, v2) = g(v2, v1), that implies that gµνis
also symmetric;
• nondegenerate: if g(v, v0) = 0∀ v, then v0= 0.
Equipped with these hypotheses, g can express the concept of infinitesimal squared distance. Furthermore, we can always find an orthonormal basis
eµ0 so as to diagonalize gµν. We choose the following convention:
g(eµ0, eν0) = 0if µ6= ν, g(e00, e00) = +1, g(ei0, ei0) = −1 (1.10)
and we will call ηµνits components.
From a physical point of view, Eq. 1.10means that we can always find a reference system such that the metric reduces to the one of flat space; but this is true only locally, for a chosen point P of the manifold. On all the manifold, it is in general not possible to reduce the metric gµν(x), defined
differently in all the tangent spaces, to a unique diagonal form.
The nondegeneracy of g can allow us to interpret the metric as a linear map v→ g(·, v) from VP into VP∗:
vµ= gµνvµ (1.11)
and, since it is an one-to-one map, the inverse also exists, defining a proce-dure to raise or lower the indices, according to convenience.
Introducing the notions of the differential calculus in curved spacetime is not trivial. A first difficulty is encountered in the definitions of derivatives: for instance, it does not make sense to consider quantities like ∆Aµ =
Aµ(P) − Aµ(Q), given a vector field Aµand two points P and Q of the
man-ifold, because Aµ(P)and Aµ(Q)belong to different vector spaces. Thus, we first need an operation that transports Aµ(Q)into the vector space of Aµ(P), such that ∆Aµtransforms simply according to the rules of the tangent space VP.
The solution is to use the covariant derivatives Dµ:
DµAν≡ ∂µAν+ Γµρν Aρ (1.12)
where, besides the usual differential ∂µAν, there is an additional piece that
allows DµAν to transform correctly; Γµρν is the connection and it is not a
tensor (its explicit expression is given in AppendixA).
Let us consider a curveC, connecting two points, P and Q, of the manifold. A vector Aµis said to be parallel transported alongC if
TµDµAν = 0 (1.13)
is satisfied in each point ofC, being Tµthe vector tangent toC in that point. This equation tells us the correct way to transport a vector from P to Q.
1.2 the einstein action 3
IfC is a closed line, it is in general not true that a parallel transported vector comes back to its original orientation, after turning around. We can define the (0, 2) type operator DµDν− DµDν that acts on covectors, producing
a (0, 3) tensor; it then can be rewritten as the action of a (1, 3) tensor on covectors as:
DµDνωρ− DνDµωρ= Rµνρσωσ (1.14)
Rµνρσ is the Riemann tensor, whose explicit expression and symmetries are
enlisted in AppendixA. Contracting the first and the third indices, we get the Ricci tensor Rµν:
Rµν= gαβRαµβν (1.15)
whose trace is the scalar curvature R:
R = gαβRαβ (1.16)
If Rµνρσ is zero, then the Ricci tensor is zero and if Rµν is zero, then the
curvature is also zero; the vice versas are not true.
1.2 t h e e i n s t e i n a c t i o n
The starting point of each quantum field theory is a classical action. In natu-ral units, it is a dimensionless scalar quantity, that in curved spacetime can be cast in the form:
Scl= Z
d4x√−gL(x) (1.17)
whereL(x), the Lagrangian density, is a local scalar function of the metric gµν and √−gis necessary to make the integral measure invariant under
changes of coordinates.
L(x) has to be chosen carefully and some conditions have to be satisfied. Besides being a scalar built only with the covariant objects listed in Sec.1.1, it should not contain second or higher order derivatives of the metric: this means that, in the equations of motion, derivatives of third or higher orders do not appear. It is essential, as we will discuss further in the following chapter, if we want to obtain stable equations of motion. Moreover, if we consider a pointlike mass as the source of the field, in the weak field limit, we would like to get the classical Newtonian equations of motions.
It is not possible to build a scalar only with the metric gµν and Γµνρ that
satisfies all these requisites, due to the nontensorial behavior of the connec-tion Γµνρ . The only scalar enlisted in Sec.1.1, that is the scalar curvature R, has the awkward feature that does contain second order derivatives, and the situation gets worse if we include contractions such as RµνRµν. Thus, since
we have no other ingredients, let us set:
Scl= Z d4x√−gL(x) = − 2 χ2 Z d4x√−g R (1.18)
where χ is a constant with dimensions −1. We will set:
4 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
where G ∼= 6.67× 10−8cm3g−1s−2is Newton’s gravitational constant.
The equations of motions are now derived by the principle of least action:
δScl= 0 (1.20)
for arbitrary variations δgµνof the metric tensor.
It is immediately clear that, with respect to the field theories defined in flat spaces, one more variation - the one of the invariant measure - appears:
δScl= − 2 χ2
Z
d4x (δ√−g) R +√−g gµν(δRµν) +√−g (δgµν)Rµν
(1.21) Let us look at the second term in 1.21. In a locally flat system, i.e. where Γµνρ = 0in a certain point P:
gµνδRµν= gµν ∂αδΓµνα − ∂νδΓµαα
(1.22) The quantities δΓα
µνdo transform like tensors: in fact, let us consider another
point P0, infinitesimally close to P and a vector Aµdefined in P0. Let us now
parallel transport it to P through two different displacements, one varying the connection, the other leaving it invariant. The difference of this two parallel transported vectors is δΓµνα Aαdxν and transforms like a vector in
VP. Thus, we can set:
gµνδRµν≡ ∂αwα (1.23)
where wµis a four vector. This is of course not true for a generic reference system, for Eq.1.23is not covariant; indeed we can guess the general result by simply substituting ∂α→ Dαin the previous result:
gµνδRµν= Dαwα=
1
√−g∂α(√−g wα) (1.24)
Therefore, when integrated inRd4x√−g, by the Gauss theorem, it reduces into an integral of wα over the surface surrounding the whole volume,
where the variations of the field are zero, and it does not contribute to the action: hence, in the equations of motion, derivatives of third or higher or-der do not appear.
Finally we get: δScl= Z d4x δL = − 2 χ2 Z d4x√−g Rµν− 1 2gµνR δgµν= 0 (1.25) or, since the variations are arbitrary:
Rµν−
1
2gµνR = 0 (1.26)
These are the well known Einstein equations - without matter - and are clearly nonlinear.
1.3 t h e g r av i t o n
In quantum field theory, the classical interactions are interpreted as ex-changes of virtual particles. The classical gravitational field, when promoted
1.3 the graviton 5
to a quantum one, is not an exception: the influence of the spacetime cur-vature on the other particles can be explained as their interaction with a specific quantum field, that we call graviton.
Thus, let us see what properties we should expect for the graviton, starting from experimental facts.
t h e g r av i t o n i s m a s s l e s s Let us suppose that the on-shell graviton has a mass µ > 0; then, we should expect the static potential between two fermions to be Yukawa-like:
ϕ∝e
−µr
r (1.27)
with a range of order 1/µ. This can be useful to set an upper bound on the graviton mass. However, in this work, we shall directly make the assumption that the graviton is massless: it produces a Newtonian potential that falls off like 1/r, without any exponential suppression. t h e g r av i t o n h a s i n t e g e r, even spin The existence of a static
poten-tial tells more about the spin of the graviton.
f f¯
t
Figure 1.1: Scattering between two spin-12 particles - f and its antiparticle ¯f - medi-ated by a virtual graviton.
Recalling that the static potential is essentially the low energy limit of a scattering process like the one represented in Fig.1.1, if its spin were half-integer, then the spins of the final particles would inevitably change and their matter structure would be different. Only integer spins can justify unaltered internal states. Moreover, an odd integer spin for the intermediate particles - just as in the case of photons - gen-erates a repulsive potential if the two particles carry the same charges and an attractive one, if the charges are different. Then, we conclude that the spin has to be even.
t h e g r av i t o n h a s s p i n 2 Now, let us suppose that the graviton is spin-less, that is to say, it is described by a scalar field. The Fourier trans-form G(k) of the propagator is then:
G(k)∝ 1
k2 (1.28)
When we compute scattering amplitudes, such as the one represented in Fig.1.1, we should contract the indices of the propagator with the indices of the stress-energy tensors of the two particles and, since in this case it does not carry any Lorentz index, the only possibility is to contract separately the indices of the stress tensors:
Tαα 1 k2T
β
6 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
Thus, a spin-0 particle can couple only with the trace of the stress-energy tensor.
The Pound-Rebka experiment in 1959 or its more recent version, the Vessot experiment in 1980, showed that the photon feels the spacetime geometry where it is moving. Its energy - thus its frequency - changes according to the intensity of the gravitational potential. If we call ϕE
the potential in the point where the photon is emitted and ϕRthe one
where it is detected, then its frequency ω changes as:
∆ω = (ϕE− ϕR)ω (1.30)
In particular, if the photon moves towards regions of lower potential (i.e. ϕE < ϕR), it experiences a redshift. From the point of view of a
quantized field theory, this is equivalent to say that the photon inter-acts with gravitons. But a spin-0 graviton would forbid such a possibil-ity, for the electromagnetic-stress tensor in1.29is traceless. Therefore, we are left with the possibility that the graviton has integer, even spin, at least 2. There are no physical reasons to discard a spin-2 graviton: hence, we suppose the graviton to be a spin-2 particle.
But we would like to represent a spin-2 particle as a convenient covariant object, that is, through a representation of the Lorentz group.
We shall consider the group SU(2)⊗ SU(2), that is homomorphic to the re-stricted Lorentz group SO(3, 1)+and classify the representations according to two spins as (s1, s2).
For example, a spin-1 particle, is described by a vector Aµ, the (12,12)
sentation. However, a four-vector is the sum of two SU(2) irreducible repre-sentations: Aµ→ 1 2, 1 2 = 1 2 ⊗ 1 2 =0⊕ 1 (1.31)
Thus, the spin-1 representation is encoded in Aµ, but it is also a redundant
representation, since it includes a spin-0 component. In addition, in the case of a massless spin-1 particle - the photon, for instance - there are only two physical degrees of freedom, against the four written in Aµ.
Following the same steps, in order to find a spin-2 representation, we can look at a rank 2 tensor τµν, which can be thought of as a dyad vµ⊗ wµand
corresponds to the sixteen-dimensional representation: τµν→ 1 2, 1 2 ⊗ 1 2, 1 2 = 1 2⊗ 1 2⊗ 1 2⊗ 1 2 (1.32)
or, in terms of irreducible representations:
τµν→ 0 ⊕ 0 ⊕ 1 ⊕ 1 ⊕ 1 ⊕ 2 (1.33)
But the tensor τµνcan be written as a sum of a symmetric part φµνand an
antisymmetric one ψµν: τµν= 1 2(τµν+ τνµ) | {z } φµν +1 2(τµν− τνµ) | {z } ψµν (1.34)
An antisymmetric tensor can always be reduced to two spin-1 representa-tions: ψ0iand εijkψjk- in the case of the electromagnetic tensor, they would
correspond to the electric and the magnetic field. Hence:
1.4 gauge fixing in the canonical formalism 7
where we have added the subscripts e and b to distinguish the two sub-spaces. Thus, from 1.33 the ten-dimensional symmetric tensor representa-tion is:
φµν→ 0w⊕ 0s⊕ 1m⊕ 2 (1.36)
where 0w denotes the spin-0 subspace φ00 (the energy), 0s is the stress
scalar φii, 1m stands for the momentum vector φ0i. Having assumed the
graviton massless, it has only two degrees of freedom and there are eight unphysical components written in φµν.
Therefore a symmetric rank two tensor φµνis a good candidate to describe
the graviton field. Actually, we already have a symmetric tensor that de-scribes the curvature of the spacetime, that is the metric gµν, but it will be
more convenient to set:
gµν≡ ηµν+ χφµν (1.37)
Being gµνdimensionless, with this definition, φµνgets the dimension of an
energy (we will explain the reason of this choice in Sec.1.5).
However, just as in the case of photon, the redundancy forces us to treat carefully a quantized version of gravity. And the clear nonlinearity of the field equations 1.26gives also a first insight of the theory we are going to deal with: quantum gravity is a gauge theory, like QED and QCD.
1.4 g au g e f i x i n g i n t h e c a n o n i c a l f o r m a l i s m
Gauge invariance is responsible of a degeneracy that makes the propagator impossible to be computed. The solution is to correct the theory by adding a gauge fixing term to the Langrangian, that breaks gauge invariance and allows us to compute the propagator. At a second stage, we must show that the physical results cannot depend on this arbitrary choice.
There are many available methods to handle gauge theories and give a sys-tematic procedure of gauge fixing; we will use a very powerful and elegant formalism developed by Batalin and Vilkovisky during the ’80s ([1], [28]
and the original paper [2]), called canonical formalism, that resembles the one
used in classical mechanics.
First, let us consider a theory described by some classical fields, that we collect into a single row:
Φicl= (φµν, Aaµ, ψ, ψ, ϕ) (1.38)
φµν represents a spin-2 field - the graviton, for instance -, Aaµ are spin-1
fields - that is to say, photons in QED or gluons in QCD -, ψ and ψ are (sets of) spinor fields and ϕ are scalar fields. In all the applications of this work, we will consider only the graviton field, and the row1.38actually reduces to the single φµν.
Let us consider a transformation of these fields, parametrized by a set of parameters Λ:
δΛΦicl= Ricl(Φcl, Λ) (1.39)
under which the action is invariant, that is: 0 = δΛScl= Z δΛΦiclδScl δΦicl = Z RiclδScl δΦicl (1.40)
8 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
The transformation in Eq.1.39satisfies the closure relation:
[δΛ, δΣ] = δ∆(Λ,Σ) (1.41)
We will see that Batalin-Vilkovisky’s canonical formalism can gather all this information into a single equation.
1.4.1 Canonical formalism
The canonical formalism requires three basic ingredients. First, once we quantize the theory, the classical fields Φi
cl are not enough and we must
enlarge them to include the ghosts, that is to say fields with fermionic statis-tics, one for each generator of the group of the transformation1.39. In our case, with the gravitational field only:
Φα= (φµν, Cµ) (1.42)
and Cµare the ghosts associated with a local coordinate transformation that
has four degrees of freedom. Now, let us define the conjugate sources:
Kα= (Kµν, KµC) (1.43)
such that their statistics are opposite to those of their respective fields: call-ing εΦα the statistics of the fields (equal to 0 if the field is bosonic, 1 if
fermionic), the statistics of the sources εKα are defined as:
εKα = εΦα+ 1mod 2 (1.44)
The last ingredient that we need is a set of functions of the fields Rα. Riare defined starting from1.39, substituting the parameter Λ with θC (where θ is a constant anticommuting parameter, while C, the ghosts, are also anticom-muting but point dependent), moving θ to the far left and finally dropping it:
θRi(Φcl, C)≡ Ricl(Φcl, θC) (1.45)
Ri(Φcl, C) can differ from Ri
cl(Φcl, θC) only by sign. For the ghost index, the
functions Rαare defined starting from the closure relation1.41, by:
RµC(θΛ + θ0Σ) = −θθ0∆(Λ, Σ) (1.46)
with θ and θ0anticommuting parameters.
Given two functionals X and Y, let us define their antiparenthesis as: (X, Y) = Z d4x δ rX δΦα δlY δKα − δrX δKα δlY δΦα (1.47) where the fields are all evaluated in the same spacetime point; δr and δl
denote respectively right and left derivatives, necessary when dealing with fermionic objects.
They satisfy the following properties:
(Y, X) = −(−1)(εX+1)(εY+1)(X, Y) (1.48)
(−1)(εX+1)(εZ+1)(X, (Y, Z)) + cyclic permutations = 0 (1.49)
If we consider a purely fermionic functional F, we trivially get:
1.4 gauge fixing in the canonical formalism 9
And for a purely bosonic functional B: (B, B) = 2 Z d4x δrB δΦα δlB δKα = −2 Z d4xδrB δKα δlB δΦα (1.51)
The nontriviality of the antiparentheses for bosons allows us to give a new definition for the quantized action.
The action S(Φ, K) is defined as solution of the master equation:
(S, S) = 0 (1.52)
with the boundary conditions: S(Φ, 0) = Scl(Φcl), − δrS(Φ, K) δKi K=0= Ri(Φcl, C) (1.53)
The minimal solution of the master equation is:
S(Φ, K) = Scl(Φcl) + SK(Φ, K) (1.54)
where Sclis the classical action and SK is defined as:
SK(Φ, K) = − Z Rα(Φ)Kα= = − Z Ri(Φcl, C)Ki− Z RCµ(Φ)KµC (1.55)
A direct consequence of the master equation and Eq.1.49is that, for each functional X:
(S, (S, X)) = 0 (1.56)
Moreover, the fields Rαcan be also seen as the antiparenthesis of the action and the fields:
Rα(Φ) = (S, Φα) (1.57)
The set of fields Rα defines the BRST transformations of the fields (after the
names of their discoverers Becchi, Rouet, Stora [3] and Tyutin [25]):
δBRSTΦα= θRα(Φ) (1.58)
where θ is anticommuting infinitesimal parameter and the nihilpotent op-erator (S,·) that generates the infinitesimal transformations is sometimes called BRST operator. We will see that they play a crucial role in the gauge fixing procedure, since they constitute the residual symmetry of the theory after applying the gauge fixing and can be used (following, in reverse, the steps that brought us to the definition of Rα) to recover the symmetry of the original theory.
Using Eq.1.51, the master equation can be rewritten as: Z d4x Rα(Φ) δlS δΦα = Z d4x Ri(Φ)δlS δΦi + RC(Φ) δlS δC = 0 (1.59)
Let us check explicitly that it condenses all the information we need. Symmetry The order zero in K of Eq.1.59gives:
Z
Ri(Φ)δlScl
10 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
that expresses exactly the invariance of the classical action under the local transformation of Eq.1.39.
Closure The first order in the sources of Eq.1.59gives: Z Rα(Φ) δl δΦα Z RβKβ= 0 ⇒ Z Rα(Φ)δlR β δΦα = 0 ∀ β (1.61)
Now let us take β = i: Z Rj(Φ) δl δΦjclR i(Φ) + Z RC δl δCR i(Φ) = 0 (1.62)
We shall now set C = θΛ + θ0Σin order to come back to a relation over the classical fields. Recalling1.39and using the linearity of R in C, the first term in Eq.1.62gives: Z θRjcl(Φcl, Λ) δl δΦjclθ 0Ri cl(Φcl, Σ) + Z θ0Rjcl(Φcl, Σ) δl δΦjclθR i cl(Φcl, Λ) (1.63) or θθ0[δΛ, δΣ]Φicl (1.64)
Since Ri are linear in C, the second term of Eq.1.62can be written as: Ricl(Φcl, RC(θΛ + θ0Σ)) = −θθ0δ∆(Λ,Σ)Φicl (1.65)
Thus, since the choice of θ and θ0 is arbitrary, Eq. 1.62 directly gives the closure relation1.41.
Jacobi identity If now, in Eq.1.61, we choose Rβ→ RC, we get: Z
RC(C)δl
δCRC(C) (1.66)
that, following the same steps as before, indeed gives the closure of the clo-sure, i.e. the Jacobi identity of the algebra.
1.4.2 Canonical transformations
A transformation of the fields
Φα0(Φ, K), Kα0(Φ, K) (1.67)
is said to be canonical if it preserves the antiparentheses, that is, for every functional X and Y:
(X0, Y0)0= (X, Y) (1.68)
where the transformed functionals X0and Y0are defined as: X0(Φ0, K0) = X(Φ(Φ0, K0), K(Φ0, K0))
1.4 gauge fixing in the canonical formalism 11
And exactly in the same way as Classical Mechanics, the canonical transfor-mations are generated by a functionalF(Φ, K0), such that:
Φα0= δF δKα0 Kα=
δF δΦα
In particular, the identity transformation is generated by the functional: I(Φ, K0) =
Z
d4x Φα(x)Kα0(x) (1.69)
It is also very useful to write a generic generating functional as the sum of a functional Ψ(Φ) dependent only on the fields Φαplus all the rest:
F(Φ, K0) = Ψ(Φ) +
Z
d4x Uα(Φ, K0)Kα0 (1.70)
1.4.3 Gauge fixing
Now we have all the necessary tools to gauge fix the theory; let us enlarge the field row of Eq.1.42so as to include antighost fields ¯Cµ- that are fields with fermionic statistics, just like Cµ, but with opposite ghost number with
respect to them - and auxiliary bosonic fields Bµ:
Φα= (φµν, Cµ, ¯Cµ, Bµ) (1.71)
adding consequently their conjugate sources ¯Kµand KBµin1.43.
The solution to the master equation can be easily obtained starting from the minimal solution to the master equation of Eq.1.54- that we now call Smin -which was obtained without ¯Cµand Bµ; it is given by:
S(Φ, K) = Smin(Φ, K) −
Z
BµK¯µ (1.72)
In the Batalin-Vilkovisky formalism the gauge fixing is a field redefinition of the kind of Eq.1.67, that is generated by a functional
F(Φ, K0) =Zd4x ΦαK0 α+ Ψ(Φ) (1.73) where Ψ(Φ) = Z d4x ¯Cµ −λ 2Bµ+Gµ(φ) (1.74) that, due to its statistical nature, is called gauge fermion andGµis the gauge
fixing function. For instance, in the harmonic gauge (the gravitational ana-logue of the Lorentz gauge of the electromagnetism) Gµ(φ) = ∂νφµν. The
transformation of the field is: Φ0α= Φα, Kα0 = Kα−
δΨ(Φ)
δΦα (1.75)
And the new, gauge fixed action SΨ(Φ, K) is:
SΨ(Φ, K) = S(Φ0, K0) = S(Φ, K) + Z
Rα(Φ)δΨ(Φ) δΦα =
12 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
that can also be written as
SΨ(Φ, K) = Scl(Φcl) + Sgf(Φ) −
Z
RαKα (1.77)
to make the gauge fixing term evident.
The fields Bµ appear only as Lagrange multipliers: they can be integrated
out or, equivalently, substituted by the solution of their own field equations. The following proposition holds:
Proposition 1.1: If the action SΨ(Φ, K) satisfies the master equation, it continues
to satisfy it even after integrating the auxiliary fields Bµout.
And, since the gauge fixing is just a particular canonical transformation: Proposition 1.2: If the action S(Φ, K) satisfies the master equation, then every SΨ= S + (S, Ψ(Φ)) satisfies the master equation.
Let us now consider a generic gauge invariant functionalQ(Φ); its expec-tation value is Z [dΦ]Q(Φ) exp iS(Φ, K) + i Z LIOI(Φcl) (1.78) where LIare other additional sources coupled to gauge invariant composite
fieldsOI, made with the classical fields. Under the canonical transformation Φα0= Φα+ θRα
Kα0 = Kα−
Z δlRβ
δΦαKβθ (1.79)
with θ an anticommuting constant, the functionalQ(Φ) and the action be-come Q(Φ0) =Q(Φ) + θZRαδQ(Φ) δΦα =Q(Φ) + θ(S, Q) S(Φ0, K0) = S(Φ, K) + θ(S, S) + Z (S, (S, Φα))Kαθ
that are exact expansions. If S satisfies the master equation, only Q(Φ) is altered by the transformation1.79; actually, Eq.1.79is just a change of vari-ables in the functional integral 1.78. In fact, the K transformation in the second line does not affect the action because S depends on K only via the combination −RRαKαand its contribution vanishes:
Z RαδlR β δΦαKβθ = Z (S, (S, Φα))Kαθ = 0 (1.80)
Therefore the extra contribution that we get from Eq. 1.78, after applying 1.79: h Z Rα δQ δΦαi 0 (1.81) (the subscript 0 recalls that the external sources coupled to the elementary fields are set to zero) has to be zero. It can be rewritten as:
1.5 quantum gravity: gravity as a gauge theory 13
and it is called Ward identity. In particular, if we chooseQ(Φ) to be the gauge fermion Ψ(Φ), we get the equivalence of the two functional measures:
Z [dΦ]exp iSΨ(Φ, K) + i Z LIOI = Z [dΦ]exp iS(Φ, K) + i Z LIOI (1.83) Therefore, more in general, the following two important theorems hold [1]:
Theorem 1.1: The correlation functions hOI1(x
1) . . .OIn(xn)i
of gauge invariant composite fieldsOIi(x
i)
• are invariant under the canonical transformation1.73, with arbitrary Ψ(Φ); • are gauge independent, i.e. their values are independent of the choice of the
gauge functionG(φ).
Theorem 1.2: The physical quantities are invariant under the most general canon-ical transformation.
1.5 q ua n t u m g r av i t y: gravity as a gauge theory
We want now to address an answer to the following issue: how can gravity be interpreted as a gauge theory? Or, equivalently, what is the local trans-formation that characterizes the theory of gravitation?
For instance, the photon free Lagrangian of QED is invariant under the transformation of the photon field
Aµ→ Aµ− ∂µΛ
where Λ is a scalar function; or, in QCD, the SU(3) local invariance makes the Lagrangian invariant under:
Aaµ→ Aaµ− DµΛa
where here Dµis the covariant derivative associated with SU(3).
If we look back at the classical gravitational action 1.18, we find that it is invariant under:
φµν→ φµν−
1
χ(Dµξν+ Dνξµ) (1.84)
where ξµ(x)is a point dependent vector. The group that generates this
sym-metry is the one of the local translations:
x0µ= xµ+ ξµ(x) (1.85)
In fact, under such a transformation, the metric tensor transforms as: gµν0 (x0) = gµν(x) − gαν∂µξα− gµα∂νξα+O(ξ2) (1.86)
But we are interested in the difference δgµν= gµν0 (x) − gµν(x)of the metric
tensors computed at the same spacetime point; then, expanding gµν0 (x0)
around x as well, we get:
14 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
that, using Eq.1.37, can be written as a difference between the fields φµν: δφµν= −ξα∂αφµν− φµα∂νξα− φαν∂µξα−
−1
χ(∂µξν+ ∂νξµ) +O(ξ
2) (1.88)
that is exactly1.84. To make the notation more compact, we can define the operator Dαµνsuch that:
δφµν= Dαµνξα (1.89)
neglecting all the termsO(ξ2).
The BRST transformations for gravity are: δBRSTφµν= θRµν(Φ) = θχ2 DαµνCα δBRSTCµ= θRCµ(Φ) = −θχ2Cν∂νCµ δBRSTC¯µ= θRµ¯ C(Φ) = θB µ δBRSTBµ= 0 (1.90)
The first line comes directly from1.89; the second line expresses the closure relation
[δΛ, δΣ]φµν= χ4(Σα(∂αΛβ) − Λα(∂αΣβ))∂βφµν
≡ δ∆(Λ,Σ)φµν (1.91)
that corresponds to:
∆µ(Λ, Σ) = χ2(Λν∂νΣµ− Σν∂νΛµ) (1.92)
To gauge fix the theory, we choose the gauge fermion: Ψ(Φ) = − 2 χ2 Z d4x ¯Cα −ξ 2Bα+ χ∂ βφ βα (1.93) The gauge fixed action is:
Sg= Scl+ Sgf (1.94) where: Sgf= − 2 χ2 Z d4x −ξ 2BµB µ+ χB µ∂λφλµ− χ3C¯µ∂νDρµνCρ (1.95) and with Scl defined in Eq. 1.18. But the field Bµ can be integrated out,
substituting it with the solution of its field equation: Bµ= χ ξ∂ νφ µν (1.96) and1.95becomes: Sgf= − 2 χ2 Z d4x χ2 2ξ∂λφ λα∂σφ σα− χ3C¯µ∂νDρµνCρ (1.97) In order to treat gravity perturbatively, we need the Feynman rules of the theory. The first feature that makes gravity peculiar is the presence of an infinite number of vertices. In fact, expanding both the curvature R and √−g
1.5 quantum gravity: gravity as a gauge theory 15
fields are generated.
The graviton propagator can be built by extracting, from the gauge fixed action, the part quadratic in the graviton field:
S(2)g =
1 2 Z
d4x φµνQµν,ρσφρσ (1.98)
The operator Qµν,ρσ has to be symmetric for exchange of the couples of
indices µν↔ ρσ and symmetric for exchanges between indices in the same couple µ ↔ ν, ρ ↔ σ. In momentum space, calling k the momentum, the propagator Pµν,ρσ(k)is defined as:
Qµν,αβ(k)Pαβ,ρσ(k) = i 2 δ ρ µδσν+ δσµδρν (1.99) where on the right side there is the identity for symmetric tensors.
In principle, Qµν,ρσcan be computed by expanding the curvature using our
field definition1.37. At this order we need: S(2)g =
Z
d4x√−g(1)R(1)+√−g(0)R(2) (1.100)
The expansions are enlisted in AppendixA, but here we do not report the result directly as it is, since it gives no particular physical insight. Indeed, it is much more useful to express Qµν,ρσusing the Barnes-Rivers spin
opera-tors [17]: looking at the group decomposition1.36, it is clear that we need four projectors on the states of definite spin. If we define the transverse and longitudinal projectors in momentum space as:
θµν= ηµν− kµkν k2 (1.101) ωµν= kµkν k2 (1.102)
the spin projectors are: Pµν,ρσ(2) = 1 2(θµρθνσ+ θµσθνρ) − 1 3θµνθρσ (1.103) Pµν,ρσ(1) = 1 2(θµρωνσ+ θµσωνρ+ θνρωµσ+ θνσωµρ) (1.104) Pµν,ρσ(0−s) = 1 3θµνθρσ (1.105) Pµν,ρσ(0−w)= ωµνωρσ (1.106)
where the dependence on the momentum is understood. They are orthonormal in the sense
Pµν(i−a)αβP(j−b)αβ,ρσ= δijδabP(j−b)µν,ρσ (1.107)
Pµν,ρσ(2) + P(1)µν,ρσ+ Pµν,ρσ(0−s)+ P(0−w)µν,ρσ = 1
2(ηµρηνσ+ ηµσηνρ) (1.108) With these projectors, the quadratic part can be expressed as:
Qµν,ρσ= k2 P(2)µν,ρσ− 2P(0−s)µν,ρσ− 1 ξ Pµν,ρσ(1) + 2Pµν,ρσ(0−w) (1.109)
16 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
whence the graviton propagator: Pµν,ρσ= i k2 " Pµν,ρσ(2) −P (0−s) µν,ρσ 2 − ξ P (1) µν,ρσ+ P(0−w)µν,ρσ 2 !# (1.110) Then the propagator falls off like∼ k−2. According to the power counting theorem (for instance, in [27]), if the propagator falls off like k−2+2s, then
the respective fields must have the dimensions of an energy to the power of 1 + s, in order to make the study of divergences with power counting work properly. That is the reason of our field definition1.37.
1.6 d i v e r g e n c e s o f q ua n t u m g r av i t y
In 1974, when t’Hooft and Veltman presented their archetypal version of quantum gravity in [23], they computed the one loop divergences of the
theory. They found that the counterterm Lagrangian is: Lcount g (gµν) = 1 16π2ε 1 120R 2+ 7 20R µνR µν (1.111) where ε = 4 − D parametrizes the divergences in the dimensional regular-ization framework and D is the continued dimension.
But they also noted that the theory is one-loop finite, that means that the coun-terterms of Eq.1.111can be eliminated by an appropriate field redefinition. Let us consider a new field gµν0 , that differs from the field gµνin1.111by δgµν:
gµν0 = gµν+ δgµν (1.112)
Expanding the gravitational Lagrangian around the field gµν, we get:
Lg(gµν0 ) =Lg(gµν) + δLg(gµν) δgµν gµν δgµν (1.113)
and, in order to cancel out the divergences, we must set: δLg(gµν) δgµν g µν δgµν≡ Lcountg (gµν) (1.114)
Let us consider a more general version of1.111: Lcount
g (gµν) = AR2+ BRµνRµν (1.115)
We can set:
δgµν= aRµν+ bRgµν (1.116)
which, substituted in1.114, gives: a = −χ 2 2 A, b = χ2 2 B +A 2 (1.117) Hence, the theory can be made finite at one loop. It was then reasonable to inquire whether the theory is finite at a higher number of loops. In 1985, Go-roff and Sagnotti [9] examined the two loops behavior of quantum gravity;
they found a counterterm of the form:
1.6 divergences of quantum gravity 17
that could not be reabsorbed by a field redefinition, making all the hopes of a finite quantum gravity fade away.
All the difficulties in renormalizing the theory come from the definition1.37 of the gravitational field. In detail, let us now consider a generic diagram D built using some of graviton vertices and propagators. It can depend on some external momenta piand the corresponding integral ID over the loop
momenta kiassumes the general form:
ID∼ Z d4k1. . . d4kL V Y i=1 V(Ni) i ({ki, pi}) YI j=1 Pj( kj, pj ) (1.119)
up to a constant symmetry factor and where we have called: • I the number of internal propagators;
• L the total number of loops (i.e. the momenta to integrate); • V the number of vertices.
They satisfy the topological relation:
L − I + V = 1 (1.120)
Moreover, V(Ni)
i is a generic Ni-leg vertex and Pj is the propagator 1.110, both in momentum space.
Let us call Λ the high energy scale: that is, each internal line carries a mo-mentum k2i → Λ2 in such a limit. Then, from Eq. 1.110, it is clear that, in
high energy regime, the graviton propagator falls off as: Pj∼ 1
Λ2 (1.121)
Since all the vertices are born from the curvature that contains two deriva-tives, going to the momentum space they behave at most like:
V(Ni)
i ∼ Λ
2
(1.122) that is, we are considering the worst case, where the legs of the vertices that carry the momenta are internal to the diagram. Thus, the integrand in1.119 falls off as Λ2(V −I)and the superficial degree of divergence is equal to:
ωD= 4L − 2I + 2V = 2(L + 1) (1.123)
A diagram can diverge if
ωD> 0 (1.124)
Now the nonrenormalizability of quantum gravity is also evident by power counting; in fact, ωDin Eq.1.123increases with the number of loops, mak-ing the renormalization process generate more and more terms to correct the divergences at each order of the perturbative expansion.
A useful theorem, stated in [26], tells what kind of counterterms we should
expect at each order:
Theorem 1.3: The leading L-loop divergences of the action1.18in four dimensions are of the form:
SdivL = χ
2(L−1)
εL
Z
d4x√−gA(x) (1.125)
18 f r o m c l a s s i c a l t o q ua n t u m g r av i t y
(i) it is a local scalar constructed from the metric tensors gµν(x);
(ii) it does not depend on χ;
(iii) it is constructed from the product of L − k + 1 Riemann tensors and 2k covari-ant derivatives Dµ, with k integer and 0 6 k 6 L.
Proof. Let us consider a generic L-loop diagram, built only with the graviton propagators1.110and vertices that stem from the action1.18. Expanding the action 1.18, using the definition of the field φµν 1.37, we see that an n-leg vertex carries χn−2. Thus, the total powers of χ that an L-loop diagram carries is: V X i=1 (ni− 2) = V X i=1 ni− 2V (1.126)
Since a propagator is connected to two internal vertex legs, calling E the number of the external, we also have the relation:
2I + E = V X i=1 ni (1.127) And1.126becomes: V X i=1 (ni− 2) = 2L − 2 + E (1.128)
Comparing this result to the exponent of χ in Eq. 1.125, we see that here we have an additional E. But the constants χE are reabsorbed when we
ex-press the diagrams through the metric tensor gµν, by means of Eq.1.37. This proves that the χ factor in Eq.1.125is correct.
The divergences can only be Lorentz scalars and the ingredients to build them are only three: the metric tensor gµν, the Riemann tensor Rµνρσ and
the covariant derivative Dµ, that can only appear in pair. Moreover, the
dimension of the scalar A(x) has to be four, in order to make SdivL dimen-sionless. Finally, the only counterterms allowed are the ones expressed in the condition (iii).
For example, at one-loop, there are only three possible counterterms:
R2, RµνRµν, RµνρσRµνρσ (1.129)
A counterterm of the form RµνρσRµνρσ cannot be reabsorbed by means
of a field redefinition of the kind of Eq. 1.112. But in four dimensions the Gauss-Bonnet theorem states that:
√−gR2− 4RµνR
µν+ RµνρσRµνρσ
=total derivative (1.130) allowing to reparametrize a divergence proportional to RµνρσR
µνρσ as a
linear combination of divergences proportional to R2 and RµνR
µν: it is this
crucial identity that makes quantum gravity one-loop finite in four dimen-sions. At two loops, more counterterms can be generated:
(DαR)(DαR), (DαRµν)(DαRµν),
R3, R RµνRµν,
RµρRνσRµνρσ, RνµRρνRµρ,
R RµνρσRµνρσ, RµνρσRµνρτRτσ,
1.6 divergences of quantum gravity 19
All these terms can be reabsorbed by a field redefinition, but the two in the last line. Moreover, RµνρσRρστζRτζµν and RµνρσRµτρζRντσζ are not
independent and one of them can be expressed in terms of the other; hence all the two loop divergences are proportional to Rµν
ρσRρστζRτζµν.
Thus, it is impossible to get just a renormalizable theory with a trivial quan-tization of the classical action 1.18. In the next chapter, we will examine how slight modifications of this action can solve these problems partially or totally.
2
T H E P R O B L E M O F
R E N O R M A L I Z A B I L I T Y
2.1 s t e l l e’s theory: higher derivative quantum gravity We saw that quantum gravity is nonrenormalizable due to the presence of the coupling constant χ with negative energy dimension, that is born from the gravitational field definition1.37, which is strictly connected with the be-havior of the propagator1.110. But if the propagator falls off faster for high values of the momentum (at least as 1/k4), then we can drop the coupling
constant χ in the change of variables of1.37, so as to deal with a dimension-less field that makes the theory renormalizable by power counting.
In 1977 Stelle proposed [19] to enlarge the gravitational action 1.18 with higher derivative terms that provide the correct UV behavior. However, the new terms that we add have to satisfy some basic requirements, such as Lorentz covariance, and we must add terms that contain parts quadratic in the graviton fields: for instance, adding terms like R3 and R4 alone are useless for our purpose, since they produce terms with at least three or four graviton fields; instead, terms like R2 or RR start quadratically in the graviton field and modify the original propagator1.110. But Stelle proposed a minimal solution: inspired by the one loop counterterms that appear in quantum gravity1.111, he added their functional form directly to the origi-nal action: SHD= − 2 χ2 Z d4x√−gR + αχ2R2+ βχ2RµνRµν (2.1)
We do not need to add a term proportional to RµνρσRµνρσ, because, in
four dimension and for spaces topologically equivalent to the Euclidean one, the Gauss-Bonnet theorem1.130holds.
The gravitational field φµνis now defined by:
gµν≡ ηµν+ φµν (2.2)
that is dimensionally correct a posteriori; in fact, let us choose the gauge fermion Ψ = − 2 χ2 Z d4x ¯Cαω − µ2 −ξ 2Bα+ ∂ βφ βα (2.3) where ω− µ2
- µ is a constant with the dimensions of an energy - is a polynomial function that, in the momentum representation, grows at least as∼ k2. It provides the gauge fixing action:
SHD, gf= − 2 χ2 1 2ξ Z d4x ∂λφλαω − µ2 ∂σφσα− − Z d4x χ3C¯µω − µ2 ∂νDρµνCρ (2.4) 21
22 t h e p r o b l e m o f r e n o r m a l i z a b i l i t y
and the quadratic operator to be inverted is QHDµν,ρσ= k 2 χ2 1 − χ2βk2P(2)µν,ρσ− − 21 + 2χ2k2(3α + β)P(0−s)µν,ρσ− −ω(k 2/µ2) ξ Pµν,ρσ(1) + 2Pµν,ρσ(0−w) (2.5) whence the propagator:
Pµν,ρσHD = iχ 2 k2 P(2) µν,ρσ 1 − βχ2k2 − P(0−s)µν,ρσ 2 1 + 2χ2k2(3α + β) − − ξ ω(k2/µ2) P (1) µν,ρσ+ P(0−w)µν,ρσ 2 ! (2.6) It falls off like∼ 1/k4in the high energy regime, justifying our definition2.2 of the field.
We also note that, not to have tachions (i.e. to have positive definite mass squared at the denominators), we have to introduce the constraints:
β > 0 3α + β < 0 (2.7) And there are some dangerous choices of the parameters: for example, the limit β → 0 cannot be taken, or there will be a part of the propagator2.6 that falls off like∼ 1/k2. Instead, the limit α→ 0 is safe.
Power counting is now very simple: using the same notation as in Sec. 1.5 and noting that the vertices behave like ∼ k4 because of the newly added higher derivative terms, the superficial degree of divergence is:
ωD = 4L − 4I + 4V = 4 (2.8)
In contrast with Eq. 1.123, it does not increase with the number of loops; moreover, all the coupling constants α, β and −2/χ2 have positive or null dimension: in sum, the theory is renormalizable by power counting. A more complete demonstration of renormalizability at each order of the perturbative series can be found in Ref. [19].
2.2 t h e l o s s o f u n i ta r i t y i n h i g h e r d e r i vat i v e m o d e l s
Although Stelle’s theory of gravitation achieves renormalizability, it shares the problems of higher derivative theories [18]. From a classical point of
view, higher derivatives introduce instabilities in the solution of the equa-tions of motion. There are some classical theories where higher derivatives naturally occur, typically seen as corrections to a lower order derivative theory. An example is the Abraham-Lorentz model for a radiating charged particle: for null external forces the equation of motion reads:
˙v − τ ¨v = 0 (2.9)
where τ =3mc2e23 ∼ 10−23s, and has two possible solutions:
˙v(t) = 0 aet/τ (2.10)
2.2 the loss of unitarity in higher derivative models 23
The same happens in its relativistic version, the Dirac equation: ˙vµ= τ ¨vµ+ ˙vν˙v ν c2 vµ (2.11) where now, beside the obvious solution vµ= 0, we also have:
vt(s) =cosh esτ+ a vx(s) =sinh eτs + a (2.12)
Both the second solution of2.9and2.12are signals of an acausal behavior. Instead, from a quantum perspective, higher derivative terms generate ghosts. In fact, let us rewrite the propagator2.6in the following way, in the Landau gauge ξ = 0: Pµν,ρσHD = iχ2 P(2) µν,ρσ k2 − P(0−s)µν,ρσ 2k2 + + P (0−s) µν,ρσ 2(k2+ 1 2(3α+β)χ2) − P (2) µν,ρσ k2− 1 βχ2 (2.13) According to Kallen-Lehmann spectral representation (for instance, in [15,
Chapter VII]), we can read the particle content of the theory examining the poles in k2of the propagators. Then, we see that the first line of2.13 corre-sponds to a massless spin-2 particle, i.e. the graviton itself, that was already present in quantum gravity1.110and it brings two degrees of freedom; the first term in the second line is associated with a spin-0 particle, with mass m20 ≡ −1/[2(3α + β)χ2] and one degree of freedom, and the last one with a spin-2 particle, whose mass is m22 ≡ 1/βχ2 and has five degrees of
free-dom. However, the last term of the propagator has the wrong sign, that is the residue at m2
2 is negative. But, since the residue is essentially a
mea-sure of the probability to create a particle (i.e.| h0| Φ(0) |ni |2, where Φ(0) is
the Heisenberg operator and |ni the eigenstate of the particle to create [15,
Chapter VII]), in these conditions unitarity cannot be achieved.
This problem is not due to our particular choice of higher derivative terms. We could have added other terms with higher derivatives, such as RcR,
R2
cR, ... Rn−1c R (where R is any contraction of the Riemann tensor), to
make the propagator fall off like ∼ 1/(k2p
n(k2)), where pn(k2) is a
poly-nomial of degree n. But applying the fundamental theorem of algebra, we could have written it as:
1 k2(1 + p n(k2)) = c0 k2 + n X i=1 ci k2− m2 i (2.14) and multiplying both the sides by k2 and taking the limit k2 → ∞, we would have got:
0 = c0+
n
X
i=1
ci (2.15)
that is to say, at least one residue would always be negative.
Therefore, even if higher derivative theories of gravitation are satisfactory from the point of view of renormalization, the price to be paid, the loss of unitarity, is too high and it does not allow to give them a direct physical interpretation.
24 t h e p r o b l e m o f r e n o r m a l i z a b i l i t y
2.3 k u z’min’s theory: nonlocal quantum gravity
From the structure of the propagator of Stelle’s theory 2.6, it is clear that unitarity can be restored (at least at the tree level) if the terms that we add to the original action 1.18do not give rise to new poles in the propagator. This can be achieved only if we abandon the locality of the Lagrangian, allowing for the presence of a transcendental function along with the usual local derivative terms. This approach was first presented by Kuz’min in a short article in 1989 [10]; it then appeared eight years later in an article by
Tomboulis [24] and recently a new interest in nonlocal field theories has
grown (for example, in [13], [14] or [5], [6], [20]).
The nonlocal action is defined by:
SNL = Z d4x√−gLNL= − 2 χ2 Z d4x√−g R + αχ2Rh1 −c µ2 R+ + βχ2Rµνh2 −c µ2 Rµν (2.16)
where µ is a constant with the dimension of a mass and specifies the en-ergy scale. The functions h1(z) and h2(z) are introduced to get a natural
generalization of Stelle’s action2.1: they can be chosen to be analytical, tran-scendental functions, that is, they can be represented as infinite series of derivatives.
The gauge fixing action is the same as Eq.2.4, but now ω (z) is a function that, in the high energy regime, grows faster than both zh1(z)and zh2(z).
Then, the quadratic operator is: QNLµν,ρσ= k 2 χ2 1 − χ2βk2h2(k2/µ2) P(2)µν,ρσ− − 21 + 2χ2k2(3αh1(k2/µ2) + βh2(k2/µ2)) Pµν,ρσ(0−s)− −ω(k 2/µ2) ξ Pµν,ρσ(1) + 2Pµν,ρσ(0−w) (2.17) from which the propagator becomes
Pµν,ρσNL = iχ 2 k2 P(2) µν,ρσ 1 − βχ2k2h 2(k2/µ2) − − P (0−s) µν,ρσ 2 1 + 2χ2k2(3αh 1(k2/µ2) + βh2(k2/µ2)) − − ξ ω(k2/µ2) P (1) µν,ρσ+ P(0−w)µν,ρσ 2 ! (2.18) With a proper choice of the functions h1(z)and h2(z)we can get rid of the
unphysical poles.
However, interpreting this theory as an extension of quantum gravity1.18, we would also like to obtain general relativity as a limit of the nonlocal action 2.16 and, in the meantime, we would also like to find the positive features of higher derivatives theory - that is, the renormalizability - in the nonlocal theories as well.
2.4 some properties of entire functions 25
formulate over the nonlocal functions h1(z)and h2(z)in order to make the
theory consistent with the previously examined ones.
2.4 s o m e p r o p e r t i e s o f e n t i r e f u n c t i o n s
Let us recall some properties of analytic functions, that we need in the fol-lowing sections. We follow the exposition in [12].
An entire function: f(z) = ∞ X k=0 akzk (2.19)
has an infinite radius of convergence and can be classified according to its behavior at the point z =∞:
• if f(z) is not singular at the point z = ∞, then, as a consequence of Liouville’s theorem, it is a constant;
• if z = ∞ is a simple pole, f(z) is a polynomial, i.e ak= 0∀ k > k0;
• if z = ∞ is an essential singularity, f(z) is said to be transcendental. Considering the results of the previous section, we will focus only on the last kind of functions.
Let us call r =|z|. Given r, we can define the maximum modulus function as: M(r) =max
|z|=r|f(z)| (2.20)
Clearly, the modulus of any transcendental function grows as r→ ∞: lim
r→+∞M(r) = +∞ (2.21)
Thus, it is not particularly illuminating to classify such functions through the limit of their modulus at z = ∞. Indeed, it is useful to compare the growth of the modulus with an exponential; that is, let us suppose that, for some µ∈ R, we have:
M(r) < erµ (2.22)
we define the order ρ of the function f(z) as:
ρ≡ inf µ (2.23)
A practical way of computing the order is given by the following formula: ρ = lim
r→+∞
ln ln M(r)
ln r (2.24)
We can also subdivide the set of functions of order 0 < ρ <∞; let us suppose that exists K > 0 such that
M(r) < eKrρ (2.25)
then, we can define the type σ of the entire function f(z) as:
26 t h e p r o b l e m o f r e n o r m a l i z a b i l i t y
and it can be computed through the formula:
σ = lim
r→+∞
ln M(r)
rρ (2.27)
For example, ez, cos z and sin z are all functions of order 1 and type 1, while eez is of infinite order.
In the following sections, we will often have to study functions inside cone-like sectors of the complex plane. There is a useful result, a simple conse-quence of Liouville’s theorem, that allows us to study the behavior of an entire function inside a conical region by simply knowing its behavior on the boundaries:
Theorem 2.1: Let Γ be the interior of an angle of π/ρ radians Γ = z α − π 2ρ <arg z < α + π 2ρ (2.28) with α∈ R and ∂Γ its boundary and let f(z) be an entire function of order ρf< ρ,
such that:
|f(z)| 6 C < ∞ ∀ z ∈ ∂Γ (2.29)
Then:
|f(z)| 6 C < ∞ ∀ z ∈ Γ (2.30)
Two corollaries, that give further information about the order of the func-tion, can be derived:
Corollary 2.1: Let us consider a family of rays emanating from the origin, such that the angle between every pair of adiacent rays does not exceed π/ρ, where ρ > 1/2. Then, every entire function f(z), that is not constant, of order ρf < ρ is
unbounded on at least one ray of the family.
Corollary 2.2: Let f(z) 6= constant be an entire function of order ρf < 1/2
or of order ρf = 1/2 and minimum type. Then f(z) is unbounded on every ray
emanating from the origin.
2.5 h y p o t h e s e s f o r t h e n o n l o c a l f u n c t i o n
Let us now discuss the properties that we must provide the nonlocal func-tions with in order to achieve not only unitarity, but also a theory consistent with general relativity and renormalizability. We shall therefore analyze the functions appearing at the denominators of the nonlocal propagators2.18:
f1(k2) = 1 − βχ2k2h2(k2/µ2)
f2(k2) = 1 + 2χ2k2(3αh1(k2/µ2) + βh2(k2/µ2))
However, throughout this work, we will uniform the discussion, concentrat-ing only on their essential functional structure:
2.5 hypotheses for the nonlocal function 27
Since h(z) is an entire function, so is f(z).
Clearly, the hypotheses that we formulate for f(z) have to be satisfied by both f1(z)and f2(z), as well as the hypotheses for h(z) have to be valid for
h2(z)and 3αh1(z) + βh2(z).
We require that:
(i) f(z) has no zeroes in the complex plane|z| < ∞; (ii) f(z) is real and positive on the real axis;
(iii) |h(z)| has the same asymptotic behavior along the real axis at ±∞; (iv) f(0) = 1;
(v) there exists 0 < Θ < π/2, such that, for the complex values of z in the conical regionC defined by:
C = {z | − Θ < arg z < Θ, π − Θ < arg z < π + Θ} (2.32) there exist γ∈ N and a real constant C such that
lim z→∞ h(z) zγ = C (2.33)
(vi) along the real axis lim
|z|→∞
h(z) − qγ(z)
qγ(z)
zm = 0, ∀ m ∈ N (2.34)
where qγ(z)is a real polynomial of degree γ.
The first hypothesis is fundamental in order not to generate new poles, in contrast with the higher derivative approach. From the theory of entire func-tions, we know that such a function can only be the exponential of another entire function g(z):
f(z)≡ eg(z) (2.35)
Condition (ii) is also necessary: the reality of f(z) assures the reality of the nonlocal action2.16and the positiveness ensures that the nonlocal function does not change the sign of the residues, letting the unitarity condition be satisfied.
The hypothesis (iii) was not present in the original Kuz’min’s paper [10],
where an asymmetric behavior of the function was allowed. Here, like in more recent papers (for instance, [14]), we prefer to adopt a symmetric
ap-proach. We also note that in Euclidean spacetime, if (v) holds, this hypothe-sis follows straightforwardly.
The hypotheses (iv) and (v) make the nonlocal theory consistent with the other previously examined. In fact, the condition (iv) states that, in the limit k2 → 0 the propagator 2.18becomes the propagator 1.110: that is, in the infrared limit, we recover general relativity, consistently with our interpreta-tion of classical gravity as an effective theory of a more complex one. Indeed, with the fifth hypothesis, in the ultraviolet regime, the nonlocal theory of gravitation resembles a higher derivative theory and therefore it is renormal-izable. Hence, the true nonlocality emerges only in a middle energy range, as depicted in Fig.2.1.
28 t h e p r o b l e m o f r e n o r m a l i z a b i l i t y
b
0 k2
k2→ 0 k2
→ +∞ General Relativity Higher derivative theory
Figure 2.1
We have imposed the convergence not only on the real axis, but also in the conical region that surrounds it.
Finally, we have also added the hypothesis (vi) (that is not present in any paper, but has been already stated in [16]): it is a subtlety regarding the UV
divergences and we will discuss it in Sec.4.5.
2.6 a n e x p l i c i t e x a m p l e o f n o n l o c a l f u n c t i o n
Obviously many functions that satisfy the hypotheses (i)-(vi) can be chosen. We will now give a very general form of functions consistent with them: we can choose g(z) defined in Eq.2.35such that:
g(z)≡ Zp(z)
0
1 − ζ(ω)
ω dω (2.36)
ζ(z)is an entire function and p(z) a real polynomial of degree γ + 1. The hypotheses (iii), (iv) and (v) can be stated as hypotheses over p(z) and ζ(z)as follows:
1. p(0) = 0;
2. ζ(z) is real and even on the real axis such that ζ(0) = 1; 3. |ζ(z)| → 0 for |z| → ∞ in the conical region defined in Eq.2.32
The hypothesis according which ζ(z) has to be even (not present in the original Kuz’min’s paper) is necessary in order to achieve the condition (iii). From corollaries 2.1 and2.2, it is clear that, if we want to build a function unbounded only in some conical regions of complex plane and bounded, in particular, in the region surrounding the real axis, we have to look for functions of order ρ > 1/2. This also implies that the function f(z) of Eq. 2.31is of infinite order.
Hence, we can choose, for example, ζ(z) as a function of order 2:
ζ(z)≡ e−z2 (2.37)
In order to study the convergence of f(z), consistently with theorem2.1, let us divide the complex plane in four cone-like sectors Sj, with a common
vertex at the origin: Sj= z −π 4 + jπ 2 <arg z 6 π 4 + jπ 2 , j = 0, 1, 2, 3 (2.38)
Then, choosing 0 < ε < π/2, we define the closed anglesSjas:
Sj= z −π 4 + jπ + ε 2 6 arg z 6 π 4 + jπ − ε 2 (2.39) andSj⊂ Sj∀ j.
Denoting θ = arg z, ζ(z) becomes:
2.6 an explicit example of nonlocal function 29
whence its modulus:
|ζ(z)| = exp(−|z|2cos 2θ)
On the boundaries of the evenSjsectors j = 0, 2, cos 2θ = sin ε, hence:
|ζ(z)| = exp(−|z|2
sin ε)|z|→∞−→ 0
Theorem2.1ensures the convergence inside the sectors as well. In the odd sectors j = 1, 3, we have cos 2θ < − sin ε and:
|ζ(z)| > exp(|z|2
sin ε)|z|→∞−→ ∞ The convergence is depicted in Fig.2.2.
θ = π/4 θ = 3π/4 θ = 5π/4 θ = 7π/4 |ζ(z)| → 0 |ζ(z)| → 0 |ζ(z)| → ∞ |ζ(z)| → ∞ π/4− ε/2
Figure 2.2: The growth of the function|ζ(z)|. The dashed lines denote the sectors Sj,
the solid ones denote the sectors ¯Sj
Hence, we could be tempted to choose the angle defining the cone region of Eq. 2.32 as Θ = π/4, but we recall from our choice of g(z) in Eq.2.36 that ζ(z) is then integrated in its variable and the right convergence has to be ensured over all the integration domain. Then, the correct choice of the angle is the one derived by our previous computation, divided by the degree of the polynomial p(z):
Θ≡ π
4(γ + 1) (2.40)
We could as well have chosen a more general definition of ζ(z) as: ζ(z)≡ exp − N X k=1 akz2k ! (2.41) with ai ∈ R and with the only restriction aN > 0, but our analysis would
have not been different. In fact, in this case, the sectors defined in Eqs.2.38, 2.39change into: Sj= z (2j − 1)π 4N <arg z 6 (2j + 1)π 4N , j = 0, . . . , 2N − 1 (2.42)
30 t h e p r o b l e m o f r e n o r m a l i z a b i l i t y Sj= z (2j − 1)π 4N + ε 2N <arg z 6 (2j + 1)π 4N − ε 2N (2.43) But we always have the convergence in half of the complex plane:
|ζ(z)||z|→∞−→ 0 j even |ζ(z)||z|→∞−→ ∞ j odd
and the angle of Eq.2.40now gets smaller by a factor N:
Θ≡ π
4N(γ + 1) (2.44)
Inserting2.37into our definition2.36of g(z), we get: g(z) = ∞ X m=1 (−1)m+1 2m m! p(z) 2m (2.45)
Recalling that the incomplete gamma function, defined by: Γ (0, x) =
Z∞
x
t−1e−tdt (2.46)
has the following Taylor expansion: Γ (0, x) = −γE−ln x − ∞ X m=1 (−z)m m m! (2.47)
where γE = 0.577 is the Euler-Mascheroni constant, we can rewrite∼ 2.45as: g(z) =1 2 h Γ0, p2(z)+ γE+ln p2(z)i (2.48) whence: f(z) = eg(z)=|p(z)|e12[Γ(0,p2(z))+γE]
It is now easy to check that the hypotheses (v) and (vi) over f(z) are satisfied. In fact, if we define: f0(z)≡ |p(z)|e γE 2 (2.49) and rewrite: f(z) = f0(z) + (f(z) − f0(z)) and use: Γ (0, x) = e−x 1 x− 1 x2+O 1 x3 (2.50) we get, on the real axis:
f(z) − f0(z) = e−p2(z) 1 2p2(z)− 1 2p4(z)+O 1 p6(z) + +Oe−p2(z) |p(z)|eγE2 −−−−→ 0|z|→∞ Hence: f(z)−−−−→ |p(z)|e|z|→∞ γE2 (2.51)
Because of the exponential suppression, the condition (vi) is also satisfied:
